题目
以下函数数中,是(x,y)=(x)^3-3x(y)^2-y的共轭调和函数的是(x,y)=(x)^3-3x(y)^2-y (x,y)=(x)^3-3x(y)^2-y(x,y)=(x)^3-3x(y)^2-y(x,y)=(x)^3-3x(y)^2-y
以下函数数中,是
的共轭调和函数的是
题目解答
答案
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解析
步骤 1：计算 $u(x,y)$ 的偏导数
首先，我们需要计算给定函数 $u(x,y)={x}^{3}-3x{y}^{2}-y$ 的偏导数。对于 $u(x,y)$，我们有：
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2$$
$$\frac{\partial u}{\partial y} = -6xy - 1$$
步骤 2：计算每个选项的偏导数
接下来，我们需要计算每个选项的偏导数，并检查它们是否满足共轭调和函数的条件。共轭调和函数的条件是：
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$
$$\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
选项 A: $v(x,y)=x+{y}^{3}+3{x}^{2}y$
$$\frac{\partial v}{\partial y} = 3y^2 + 3x^2$$
$$\frac{\partial v}{\partial x} = 1 + 6xy$$
选项 B: $v(x,y)={x}^{2}+{y}^{3}+3{x}^{2}y$
$$\frac{\partial v}{\partial y} = 3y^2 + 3x^2$$
$$\frac{\partial v}{\partial x} = 2x + 6xy$$
选项 C: $v(x,y)=2+x-{y}^{3}+3{x}^{2}y$
$$\frac{\partial v}{\partial y} = -3y^2 + 3x^2$$
$$\frac{\partial v}{\partial x} = 1 + 6xy$$
选项 D: $v(x,y)=x+{y}^{3}+3xy-1$
$$\frac{\partial v}{\partial y} = 3y^2 + 3x$$
$$\frac{\partial v}{\partial x} = 1 + 3y$$
步骤 3：验证共轭调和函数的条件
现在，我们检查每个选项是否满足共轭调和函数的条件。
选项 A:
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2 \neq \frac{\partial v}{\partial y} = 3y^2 + 3x^2$$
$$\frac{\partial u}{\partial y} = -6xy - 1 \neq -\frac{\partial v}{\partial x} = -1 - 6xy$$
选项 B:
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2 \neq \frac{\partial v}{\partial y} = 3y^2 + 3x^2$$
$$\frac{\partial u}{\partial y} = -6xy - 1 \neq -\frac{\partial v}{\partial x} = -2x - 6xy$$
选项 C:
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2 \neq \frac{\partial v}{\partial y} = -3y^2 + 3x^2$$
$$\frac{\partial u}{\partial y} = -6xy - 1 \neq -\frac{\partial v}{\partial x} = -1 - 6xy$$
选项 D:
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2 \neq \frac{\partial v}{\partial y} = 3y^2 + 3x$$
$$\frac{\partial u}{\partial y} = -6xy - 1 \neq -\frac{\partial v}{\partial x} = -1 - 3y$$
首先，我们需要计算给定函数 $u(x,y)={x}^{3}-3x{y}^{2}-y$ 的偏导数。对于 $u(x,y)$，我们有：
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2$$
$$\frac{\partial u}{\partial y} = -6xy - 1$$
步骤 2：计算每个选项的偏导数
接下来，我们需要计算每个选项的偏导数，并检查它们是否满足共轭调和函数的条件。共轭调和函数的条件是：
$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$$
$$\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$
选项 A: $v(x,y)=x+{y}^{3}+3{x}^{2}y$
$$\frac{\partial v}{\partial y} = 3y^2 + 3x^2$$
$$\frac{\partial v}{\partial x} = 1 + 6xy$$
选项 B: $v(x,y)={x}^{2}+{y}^{3}+3{x}^{2}y$
$$\frac{\partial v}{\partial y} = 3y^2 + 3x^2$$
$$\frac{\partial v}{\partial x} = 2x + 6xy$$
选项 C: $v(x,y)=2+x-{y}^{3}+3{x}^{2}y$
$$\frac{\partial v}{\partial y} = -3y^2 + 3x^2$$
$$\frac{\partial v}{\partial x} = 1 + 6xy$$
选项 D: $v(x,y)=x+{y}^{3}+3xy-1$
$$\frac{\partial v}{\partial y} = 3y^2 + 3x$$
$$\frac{\partial v}{\partial x} = 1 + 3y$$
步骤 3：验证共轭调和函数的条件
现在，我们检查每个选项是否满足共轭调和函数的条件。
选项 A:
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2 \neq \frac{\partial v}{\partial y} = 3y^2 + 3x^2$$
$$\frac{\partial u}{\partial y} = -6xy - 1 \neq -\frac{\partial v}{\partial x} = -1 - 6xy$$
选项 B:
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2 \neq \frac{\partial v}{\partial y} = 3y^2 + 3x^2$$
$$\frac{\partial u}{\partial y} = -6xy - 1 \neq -\frac{\partial v}{\partial x} = -2x - 6xy$$
选项 C:
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2 \neq \frac{\partial v}{\partial y} = -3y^2 + 3x^2$$
$$\frac{\partial u}{\partial y} = -6xy - 1 \neq -\frac{\partial v}{\partial x} = -1 - 6xy$$
选项 D:
$$\frac{\partial u}{\partial x} = 3x^2 - 3y^2 \neq \frac{\partial v}{\partial y} = 3y^2 + 3x$$
$$\frac{\partial u}{\partial y} = -6xy - 1 \neq -\frac{\partial v}{\partial x} = -1 - 3y$$